Monoid Category

A monoid category is a category with a single object whose morphisms form a monoid under composition.

Put differently, every monoid can be interpreted as a single-object category, where each element of the monoid corresponds to a morphism.

This idea creates a direct bridge between algebra and category theory. The algebraic structure of the monoid becomes the categorical structure of morphisms and their composition.

The main features of a monoid category are the following:

• A single object: The category contains exactly one object, usually denoted by the symbol \(\ast\). Every morphism starts and ends at this same object.

  Note. The specific nature of the object is not important because it is treated as an abstract entity. To build intuition, however, it may help to imagine the unique object as a single apple, a point, or any other placeholder object.

• Morphisms associated with monoid elements: Each element of the monoid is represented by a morphism from \(\ast\) to \(\ast\). Consequently, there is a one-to-one correspondence between monoid elements and morphisms in the category.
• Composition determined by the monoid operation: Morphism composition follows the binary operation of the monoid. If \( f \) and \( g \) correspond to the elements \( m \) and \( n \), then the composition \( f \circ g \) corresponds to the product \( m \cdot n \).

  Note. Since every monoid is associative by definition, morphism composition is automatically associative: \[ (f \circ g) \circ h = f \circ (g \circ h). \]

• An identity morphism: Every monoid contains an identity element, and this element becomes the identity morphism of the category. If \( e \) is the identity element, then the corresponding morphism \( \text{id}_{\ast} \) satisfies \[ \text{id}_{\ast} \circ f = f \circ \text{id}_{\ast} = f \] for every morphism \( f \).

  Note. For example, zero is the identity element in the monoid \((\mathbb{N}_0,+)\) of natural numbers with zero under addition.

The categorical structure of a monoid category mirrors the algebraic structure of the underlying monoid.

Associativity in the monoid becomes associativity of morphism composition, while the identity element becomes the identity morphism of the category.

Note. Monoid categories appear in several areas of mathematics and theoretical computer science, especially in the semantics of programming languages and in the study of algebraic structures.

A practical example

Consider the monoid of natural numbers with zero \( \mathbb{N}_0 \) under addition:

$$ (\mathbb{N}_0, + ) $$

This monoid defines a single-object category, represented by the symbol \(\ast\).

The object itself remains abstract, so its concrete interpretation does not matter.

Note. To make the example more intuitive, imagine the object as an apple placed inside a container.

In this category, the morphisms are the numbers \(0,1,2,\dots\). Each number represents the operation of adding that many objects to the container.

For example:
0 means "add zero apples"
1 means "add one apple"
2 means "add two apples"
and so on.

The composition of two morphisms \(a\) and \(b\) is given by their sum:

$$ a \circ b = a+b $$

The result is still a natural number with zero, so the operation remains inside the same set.

For instance, if \(a=2\) and \(b=5\), then \[ a+b=2+5=7. \] The result \(7\) is another morphism in the category.

The identity morphism is represented by the number \(0\).

Indeed, composing any morphism with \(0\) leaves it unchanged: \[ a+0=0+a=a. \] For example, \[ 2+0=0+2=2. \]

This example clearly shows how a purely algebraic structure can be reformulated in categorical language.

The difference between a monoid category and a groupoid

A monoid category and a groupoid describe different categorical ideas.

• A monoid category is a category with a single object whose morphisms form a monoid under composition.
• A groupoid is a category in which every morphism is invertible.

Thus, a monoid category focuses on the algebraic structure induced by composition, whereas a groupoid focuses on reversibility and invertibility.

Note. The two notions are not incompatible. If the original monoid is actually a group, then the associated single-object category is also a groupoid, because every morphism has an inverse.

The difference between a monoid category and a monoidal category

The expressions monoid category and monoidal category are often confused, but they refer to different concepts.

A monoid category arises when a monoid is represented as a category with a single object.

In this construction:

• each monoid element becomes a morphism
• composition follows the monoid operation
• the identity element becomes the identity morphism

A monoidal category, by contrast, is a much broader structure in category theory.

A monoidal category is equipped with a tensor product, which combines objects and morphisms, together with a unit object. These structures satisfy associativity and identity conditions expressed through natural isomorphisms and coherence laws.

In summary:

• A monoid category is a single-object category whose morphisms form a monoid.
• A monoidal category is a category equipped with a tensor product and a unit object.

The two concepts are related, but they are not equivalent.